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A nonconvex dissipative system and its applications (II)

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Abstract

This paper presents exact solutions in terms of implicit functions and hyperbolic functions to a nonconvex dissipative system, controlled by a Duffing–van der Pol nonlinear equation with a fifth-order nonlinearity. Applications to the complex Ginzburg–Landau equation are illustrated and several classes of uniformly translating solutions are obtained accordingly.

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References

  1. Jordan D.W., Smith P. (1977). Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford

    Google Scholar 

  2. Brand H.R., Deissler R.J. (1989). Interaction of localized solutions for subcritical bifurcations. Phys. Rev. Lett. 63: 2801–2804

    Article  Google Scholar 

  3. Kolodner P. (1992). Extended states of nonlinear traveling-wave convection (I)-the Eckhaus instability. Phys. Rev. E 46: 6431–6451

    Google Scholar 

  4. Feng, Z.: Qualitative analysis to a nonconvex dissipative system, preceding paper. J. Glob. Optim.

  5. Lu Q.S. (1992). Qualitative Theory of Ordinary Differential Equations. Beihang Univesity Press, Beijing

    Google Scholar 

  6. Gao D.Y. (2000). Analytic solutions and triality theory for nonconvex and nonsmooth variational problems with applications. Nonlinear Anal. 42: 1161–1193

    Article  Google Scholar 

  7. Gao D.Y. (2001). Nonsmooth and nonconvex dynamics: duality, polarity and complementary extremum principles, nonsmooth/nonconvex mechanics. In: Gao, D.Y., Ogden, R.W., Stavroulakis, G.E. (eds) Modeling, Analysis and Numerical Methods., pp. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  8. Luo D., Wang X., Zhu D., Han M. (1997). Bifurcation Theory and Methods of Dynamical Systems. World Scientific Publishing Co. Pte. Ltd., Singapore

    Google Scholar 

  9. Chow S.N., Hale J.K. (1982). Methods of Bifurcation Theory. Springer, New York

    Google Scholar 

  10. Guckenheimer J., Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York

    Google Scholar 

  11. Zabusky N.J., Kruskal M.D. (1965). Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15: 240–243

    Article  Google Scholar 

  12. Zabusky N.J., Galvin C.J. (1971). Shallow water waves, the Korteweg-de Vries equation and solitons. J. Fluid Mech. 47: 811–824

    Article  Google Scholar 

  13. Kruskal, M.D.: Nonlinear wave equations. In: Moser J. (ed.) Dynamical Systems, Theory and Applications. Lecture Notes in Physics, vol. 38, pp. 310–354. Springer-Verlag, Heidelberg (1975)

  14. Whitham G.B. (1974). Linear and Nonlinear Waves. Springer-Verlag, New York

    Google Scholar 

  15. Kruskal MD (1974). The Korteweg-de Vries Equation and Related Evolution Equations. American Mathematical Society, Providence RI

    Google Scholar 

  16. Ablowitz M.J., Segur H. (1981). Solitons and the Inverse Scattering Transform. SIAM, Philadelphia

    Google Scholar 

  17. Nielsen M.A., Chuang I.L. (2000). Quantum Computation and Quantum Information. Cambridge University Press, New York

    Google Scholar 

  18. Dey B. (1986). Domain wall solutions of KdV like equations with higher order nonlinearity. J. Phys. A (Math. Gen.) 19: L9–L12

    Article  Google Scholar 

  19. Fan E.G., Zhang J., Hon B.Y. (2001). A new complex line soliton for the two-dimensional KdV-Burgers equation. Phys. Lett. A. 291: 376–380

    Article  Google Scholar 

  20. Yan Z.Y. (2001). New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys. Lett. A. 292: 100–106

    Article  Google Scholar 

  21. Li B., Chen Y., Zhang H.Q. (2002). Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdVBurgers-type equations with nonlinear terms of any order. J. Phys. A (Math. Gen.) 35: 8253–8265

    Article  Google Scholar 

  22. Lu C., Fan E.G. (2001). Soliton solutions for the new complex version of a coupled KdV equation and a coupled MKdV equation. Phys. Lett. A 285: 373–376

    Article  Google Scholar 

  23. Wu, W.: Algorithms and computation. In: Du, D.Z., Zhang X.S. (eds.) Proceedings of the Fifth International Symposium (ISAAC’ 94). Lecture Notes in Computer Science, vol. 834. Springer-Verlag, Berlin (1994)

  24. Kolodner P., Bensimon D., Surko C.M. (1988). Traveling-wave convection in an annulus. Phys. Rev. Lett. 60: 1723–1726

    Article  Google Scholar 

  25. Bensimon D., Kolodner P., Surko C.M. (1990). Competing and coexisting dynamic states of traveling-wave convection in annulus. J. Fluid. Mech. 217: 441–467

    Article  Google Scholar 

  26. Wu J., Wheatley J., Putterman S. (1987). Observation of envelope solitons in solids. Phys. Rev. Lett. 59: 2744–2774

    Article  Google Scholar 

  27. Brand H.R., Lomdahl P.S., Newell A.C. (1986). Evolution of the order parameter in situations with broken rotational symmetry. Phys. Lett. A 118: 67–73

    Article  Google Scholar 

  28. Brand H.R., Lomdahl P.S., Newell A.C. (1986). Benjamin-Feir Turbulence in convective binary fluid mixtures. Phys. D 23: 345–361

    Article  Google Scholar 

  29. Hasegawa A. (1989). Optical Solitons in Fibers. Springer, New York

    Google Scholar 

  30. Kolodner P. (1992). Extended states of nonlinear traveling-wave convection (II)-the Eckhaus instability. Phys. Rev. E 46: 6452–6471

    Google Scholar 

  31. Hohenberg P.C., Saarloos W. (1992). Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations. Phys. D 56: 303–367

    Article  Google Scholar 

  32. Clarkson P.A., LeVeque R.J., Saxton R. (1986). Solitary-wave interactions in elastic rods. Stud. Appl. Math. 75: 95–121

    Google Scholar 

  33. Zhang W.G., Chang Q.S., Fan E.G. (2003). Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order. J. Math. Anal. Appl. 287: 1–18

    Article  Google Scholar 

  34. Feng Z., Ji W.D., Fei S.M. (1997). Exact solutions of a generalized Klein-Gordon equation. Math. Prac. Theory 27: 222–226

    Google Scholar 

  35. Porubov A.V., Velarde M.G. (2002). Strain kinks in an elastic rod embedded in a viscoelastic medium. Wave Motion 35: 189–204

    Article  Google Scholar 

  36. Kundu A. (1984). Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations. J. Math. Phys. 25: 3433–3438

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Texas-Pan American, Edinburg, TX, 78541, USA

    Zhaosheng Feng

  2. Department of Mathematics, Virginia Tech. University, Blacksburg, VA, 24061, USA

    David Y. Gao

Authors
  1. Zhaosheng Feng
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  2. David Y. Gao
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Correspondence to Zhaosheng Feng.

Additional information

Part of the work was announced at the International Conference on Complementarity, Duality, and Global Optimization in Science and Engineering, Virginia Tech. University, Blacksburg, Virginia, August 15–17, 2005. This work is supported by NSF Grant CCF–0514768.

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Feng, Z., Gao, D.Y. A nonconvex dissipative system and its applications (II). J Glob Optim 40, 637–651 (2008). https://doi.org/10.1007/s10898-006-9114-0

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  • DOI: https://doi.org/10.1007/s10898-006-9114-0

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